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We introduce a new definition of topological degree for a meaningful class of operators which need not be continuous. Subsequently, we derive a number of fixed point theorems for such operators. As an application, we deduce a new existence result for first-order ODEs with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

We prove an existence result for systems of differential inclusions driven by multivalued mappings which need not assume closed or convex values everywhere, and need not be semicontinuous everywhere. Moreover, we consider differentiation with respect to a nondecreasing function, thus covering discrete, continuous and impulsive problems under a unique formulation. We emphasize that our existence result appears to be new even when the derivator is the identity, i.e. when derivatives are considered in the usual sense. We also apply our existence theorem for inclusions to derive a new existence result for discontinuous Stieltjes differential equations. Examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

We present new results on existence and uniqueness of absolutely continuous solutions for systems of discontinuous ordinary differential equations. Our existence result complements an earlier theorem by Bressan and Shen. Basically, we show that a global transversality condition assumed by Bressan and Shen need only be imposed on the sets where the nonlinear part is discontinuous. Our proof, completely different to the one given by Bressan and Shen, uses Krasovskij solutions as a first step. We illustrate the applicability of our result with several examples not covered by the previous literature. The second part of this paper concerns uniqueness. Specifically, we prove uniqueness of solutions for discontinuous systems of differential equations with piecewise Lipschitz continuous nonlinearities and assuming localized Bressan–Shen transversality conditions on the boundaries between different Lipschitz continuity domains. Our uniqueness result appears to be new even in the classical case of continuous nonlinearities. [ABSTRACT FROM AUTHOR]

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel'skiĭ's fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory. [ABSTRACT FROM AUTHOR]